Past exam of the mathematics course of the University of Cambridge / 2009 / ia / Paper 2 / 3F (Probability)

With ${\sigma }_1,{\sigma }_2>0$ and $|\rho |<1$, the bivariate normal distribution has probability density functionThe strict correlation bound makes its covariance matrix positive definite, so this is an ordinary two-dimensional density rather than a distribution supported on a line.

If $\rho =0$, the bivariate normal density in part (a) factorizes asIntegrating over any product of measurable sets proves independence of random variables. Conversely, independent variables with finite second moments have $\mathbb{E}[X_1{X}_2]=\mathbb{E}[X_1]\mathbb{E}[X_2]$, so their covariance is zero and $\rho =\mathrm{Cov}(X_1,X_2)/({\sigma }_1{\sigma }_2)=0$. This proves independence of uncorrelated jointly normal variables; zero correlation alone would not imply independence without the joint normal assumption.